\int

**marijakopljar** · January 17th 2013, 08:43 AM

Hello!
Please, any good soul to help me with:
$\intx^2sin2xdx$ ???
The result is :
$\frac{1}{4}(1-2x^2)cos2x+\frac{1}{2}xsin2x+C$

Many thanks!!!

**HallsofIvy** · January 17th 2013, 09:42 AM

Originally Posted by marijakopljar

Hello!
Please, any good soul to help me with:
$\int x^2sin2xdx$ ???

You left out the space between 'int' and 'x^2'. I added it.

Use "integration by parts": $\int u dv= uv- \int v du$ with $u= x^2$ , $dv= sin(2x)dx$ so that $du= 2xdx$ , $v= -\frac{1}{2}cos(2x)$ . Doing that, you will still have an integral involving $xcos(2x)$ . So use integration by parts again, this time letting $u= x$ , $dv= cos(2x)dx$ .

Generally speaking, any time you have an integral of $x^n f(x)dx$ , you can consider repeated integration by parts, repeatedly taking the power of x as "u" so that when you differentiate to get du, you have one less power of x. Repeat until you have reduced the power of x to 0.

The result is :
$\frac{1}{4}(1-2x^2)cos2x+\frac{1}{2}xsin2x+C$

Many thanks!!!

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